Let the \(V\) -Newton polygon of \(g(x)\)  of second order has \(k\) -sides \(E_1, \cdots , E_k\)  with negative slopes \(\lambda _1, \cdots , \lambda _k\) . Let \(\lambda _t = -\frac{\ell _t}{e_t}\)  with \(\gcd (\ell _t, e_t) = 1\)  and \([a_t, b_t]\)  denote the projection to the horizontal axis of the side of slope \(\lambda _t\)  for \(1\le t\le k\) . Then, there is a natural residual polynomial \(T_t(Y)\)  of second order attached to each side \(E_t\) , whose degree coincides with the degree of the side (i.e. \(\frac{b_t-a_t}{e_t}\) ) [1]}. Only those integral points of the \(V\) -Newton polygon of \(g(x)\)  which lie on the side, determine a nonzero coefficient of this second order residual polynomial, and this is the only property we need in what follows. We define \(f(x)\)  to be \(T_t\) -regular when the second order residual polynomial \(T_t(Y)\)  attached to the side \(E_t\)  of the \(V\) -Newton polygon of \(g(x)\)  of second order is separable in \(\mbox{$\mathbb {F}$}_p[Y]/\langle \psi (Y)\rangle \) . We define \(f(x)\)  to be \(V\) -regular if \(f(x)\)  is \(T_t\) -regular for each \(t\) , \(1\le t\le k\) .
